Cook the rice according to the directions on the package. When done set aside.
Preheat oven to 375 degrees F.
In a large skillet add the butter and chopped onions. Cook over medium heat until the onions are tender about 5 minutes.
Add flour and stir in, cook for 30 seconds to 1 minute.
Add the milk and chicken broth, increase the heat slightly, and cook until the sauce starts to thicken. Once the sauce starts to thicken reduce heat back to medium.
Add 2 cups of the cheddar cheese and sour cream and stir until the cheese is melted and the sour cream is mixed in.
Add the cooked chicken, cooked rice, and broccoli and stir in. Transfer to a 9 by 13-inch baking dish.
Place in the preheated oven and bake for 25 to 35 minutes. The sides of the casserole should be bubbling.